∫ 2pi

We will calculate mean values of characteristic polynomials of matrices from U(N). A major motivation for this is applications to problems in number theory, but in fact characteristic polynomials are fundamental quantities in the study of matrices and there has been much work done on them independent of any connection to number theory. We will look at the simplest case of moments of characteristic polynomials of matrices from U(N), but this can be generalised to mean values of ratios of characteristic polynomials, autocorrelation functions and to other ensembles of random matrices. We write the characteristic polynomial of A, an N ×N unitary matrix, as ΛA(s) = det(I −A∗s) = N∏ n=1 1 − se−iθn, (1) where the eiθn are the eigenvalues of A and A ∗ is the conjugate transpose of A. We want to calculate moments of |ΛA(eiθ)|. Since this is a function of the eigenvalues, an average of |ΛA(eiθ)|s (the sth moment) can be performed over the group U(N) with respect to Haar measure using the Weyl integration formula: U(N